LMFDB - Embedded newform 1792.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1792,4,Mod(1,1792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1792.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1792.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,-34,0,-14,0,98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$105.731422730$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 448)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1792.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.66025 q^{3} -22.1962 q^{5} -7.00000 q^{7} +31.6795 q^{9} -40.1051 q^{11} -14.4833 q^{13} +170.028 q^{15} -111.569 q^{17} -117.804 q^{19} +53.6218 q^{21} +119.751 q^{23} +367.669 q^{25} -35.8461 q^{27} -92.8897 q^{29} -181.569 q^{31} +307.215 q^{33} +155.373 q^{35} +247.090 q^{37} +110.946 q^{39} -58.1333 q^{41} +249.885 q^{43} -703.163 q^{45} +101.913 q^{47} +49.0000 q^{49} +854.649 q^{51} -570.697 q^{53} +890.179 q^{55} +902.407 q^{57} +277.391 q^{59} +82.9526 q^{61} -221.756 q^{63} +321.474 q^{65} +397.969 q^{67} -917.325 q^{69} +6.22055 q^{71} +209.769 q^{73} -2816.44 q^{75} +280.736 q^{77} +271.902 q^{79} -580.756 q^{81} +1010.28 q^{83} +2476.41 q^{85} +711.559 q^{87} +730.420 q^{89} +101.383 q^{91} +1390.87 q^{93} +2614.79 q^{95} +707.451 q^{97} -1270.51 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 34 q^{5} - 14 q^{7} + 98 q^{9} - 4 q^{11} - 74 q^{13} + 56 q^{15} - 140 q^{17} - 246 q^{19} - 14 q^{21} + 288 q^{23} + 382 q^{25} + 344 q^{27} - 68 q^{29} - 280 q^{31} + 656 q^{33} + 238 q^{35}+ \cdots + 1124 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 34 * q^5 - 14 * q^7 + 98 * q^9 - 4 * q^11 - 74 * q^13 + 56 * q^15 - 140 * q^17 - 246 * q^19 - 14 * q^21 + 288 * q^23 + 382 * q^25 + 344 * q^27 - 68 * q^29 - 280 * q^31 + 656 * q^33 + 238 * q^35 - 164 * q^37 - 464 * q^39 + 244 * q^41 + 188 * q^43 - 1486 * q^45 - 392 * q^47 + 98 * q^49 + 580 * q^51 - 504 * q^53 + 464 * q^55 - 336 * q^57 - 190 * q^59 + 398 * q^61 - 686 * q^63 + 1024 * q^65 - 368 * q^67 + 708 * q^69 + 248 * q^71 - 204 * q^73 - 2678 * q^75 + 28 * q^77 - 440 * q^79 + 1298 * q^81 + 1158 * q^83 + 2812 * q^85 + 952 * q^87 + 1156 * q^89 + 518 * q^91 + 440 * q^93 + 4128 * q^95 - 428 * q^97 + 1124 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−7.66025	−1.47422	−0.737108	−	0.675775i	$-0.763809\pi$
$3$	−7.66025	−1.47422	−0.737108	+	0.675775i	$0.763809\pi$
$4$	0	0
$5$	−22.1962	−1.98528	−0.992642	−	0.121085i	$-0.961363\pi$
$5$	−22.1962	−1.98528	−0.992642	+	0.121085i	$0.961363\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	31.6795	1.17331
$10$	0	0
$11$	−40.1051	−1.09929	−0.549643	−	0.835400i	$-0.685236\pi$
$11$	−40.1051	−1.09929	−0.549643	+	0.835400i	$0.685236\pi$
$12$	0	0
$13$	−14.4833	−0.308997	−0.154498	−	0.987993i	$-0.549376\pi$
$13$	−14.4833	−0.308997	−0.154498	+	0.987993i	$0.549376\pi$
$14$	0	0
$15$	170.028	2.92674
$16$	0	0
$17$	−111.569	−1.59174	−0.795868	−	0.605470i	$-0.792985\pi$
$17$	−111.569	−1.59174	−0.795868	+	0.605470i	$0.792985\pi$
$18$	0	0
$19$	−117.804	−1.42242	−0.711212	−	0.702978i	$-0.751853\pi$
$19$	−117.804	−1.42242	−0.711212	+	0.702978i	$0.751853\pi$
$20$	0	0
$21$	53.6218	0.557201
$22$	0	0
$23$	119.751	1.08565	0.542823	−	0.839847i	$-0.317355\pi$
$23$	119.751	1.08565	0.542823	+	0.839847i	$0.317355\pi$
$24$	0	0
$25$	367.669	2.94135
$26$	0	0
$27$	−35.8461	−0.255503
$28$	0	0
$29$	−92.8897	−0.594800	−0.297400	−	0.954753i	$-0.596119\pi$
$29$	−92.8897	−0.594800	−0.297400	+	0.954753i	$0.596119\pi$
$30$	0	0
$31$	−181.569	−1.05196	−0.525981	−	0.850497i	$-0.676302\pi$
$31$	−181.569	−1.05196	−0.525981	+	0.850497i	$0.676302\pi$
$32$	0	0
$33$	307.215	1.62059
$34$	0	0
$35$	155.373	0.750367
$36$	0	0
$37$	247.090	1.09787	0.548936	−	0.835864i	$-0.315033\pi$
$37$	247.090	1.09787	0.548936	+	0.835864i	$0.315033\pi$
$38$	0	0
$39$	110.946	0.455528
$40$	0	0
$41$	−58.1333	−0.221436	−0.110718	−	0.993852i	$-0.535315\pi$
$41$	−58.1333	−0.221436	−0.110718	+	0.993852i	$0.535315\pi$
$42$	0	0
$43$	249.885	0.886210	0.443105	−	0.896470i	$-0.353877\pi$
$43$	249.885	0.886210	0.443105	+	0.896470i	$0.353877\pi$
$44$	0	0
$45$	−703.163	−2.32936
$46$	0	0
$47$	101.913	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$47$	101.913	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	854.649	2.34656
$52$	0	0
$53$	−570.697	−1.47908	−0.739541	−	0.673112i	$-0.764957\pi$
$53$	−570.697	−1.47908	−0.739541	+	0.673112i	$0.764957\pi$
$54$	0	0
$55$	890.179	2.18240
$56$	0	0
$57$	902.407	2.09696
$58$	0	0
$59$	277.391	0.612089	0.306044	−	0.952017i	$-0.400994\pi$
$59$	277.391	0.612089	0.306044	+	0.952017i	$0.400994\pi$
$60$	0	0
$61$	82.9526	0.174115	0.0870573	−	0.996203i	$-0.472254\pi$
$61$	82.9526	0.174115	0.0870573	+	0.996203i	$0.472254\pi$
$62$	0	0
$63$	−221.756	−0.443471
$64$	0	0
$65$	321.474	0.613446
$66$	0	0
$67$	397.969	0.725667	0.362833	−	0.931854i	$-0.381809\pi$
$67$	397.969	0.725667	0.362833	+	0.931854i	$0.381809\pi$
$68$	0	0
$69$	−917.325	−1.60048
$70$	0	0
$71$	6.22055	0.0103978	0.00519889	−	0.999986i	$-0.498345\pi$
$71$	6.22055	0.0103978	0.00519889	+	0.999986i	$0.498345\pi$
$72$	0	0
$73$	209.769	0.336324	0.168162	−	0.985759i	$-0.446217\pi$
$73$	209.769	0.336324	0.168162	+	0.985759i	$0.446217\pi$
$74$	0	0
$75$	−2816.44	−4.33619
$76$	0	0
$77$	280.736	0.415491
$78$	0	0
$79$	271.902	0.387233	0.193617	−	0.981077i	$-0.437978\pi$
$79$	271.902	0.387233	0.193617	+	0.981077i	$0.437978\pi$
$80$	0	0
$81$	−580.756	−0.796648
$82$	0	0
$83$	1010.28	1.33606	0.668028	−	0.744136i	$-0.267138\pi$
$83$	1010.28	1.33606	0.668028	+	0.744136i	$0.267138\pi$
$84$	0	0
$85$	2476.41	3.16005
$86$	0	0
$87$	711.559	0.876863
$88$	0	0
$89$	730.420	0.869937	0.434969	−	0.900446i	$-0.356759\pi$
$89$	730.420	0.869937	0.434969	+	0.900446i	$0.356759\pi$
$90$	0	0
$91$	101.383	0.116790
$92$	0	0
$93$	1390.87	1.55082
$94$	0	0
$95$	2614.79	2.82392
$96$	0	0
$97$	707.451	0.740523	0.370262	−	0.928927i	$-0.379268\pi$
$97$	707.451	0.740523	0.370262	+	0.928927i	$0.379268\pi$
$98$	0	0
$99$	−1270.51	−1.28981
$100$	0	0
$101$	−596.558	−0.587720	−0.293860	−	0.955849i	$-0.594940\pi$
$101$	−596.558	−0.587720	−0.293860	+	0.955849i	$0.594940\pi$
$102$	0	0
$103$	539.959	0.516541	0.258270	−	0.966073i	$-0.416847\pi$
$103$	539.959	0.516541	0.258270	+	0.966073i	$0.416847\pi$
$104$	0	0
$105$	−1190.20	−1.10620
$106$	0	0
$107$	660.497	0.596754	0.298377	−	0.954448i	$-0.403555\pi$
$107$	660.497	0.596754	0.298377	+	0.954448i	$0.403555\pi$
$108$	0	0
$109$	234.157	0.205763	0.102881	−	0.994694i	$-0.467194\pi$
$109$	234.157	0.205763	0.102881	+	0.994694i	$0.467194\pi$
$110$	0	0
$111$	−1892.77	−1.61850
$112$	0	0
$113$	−1740.74	−1.44916	−0.724578	−	0.689193i	$-0.757965\pi$
$113$	−1740.74	−1.44916	−0.724578	+	0.689193i	$0.757965\pi$
$114$	0	0
$115$	−2658.02	−2.15532
$116$	0	0
$117$	−458.825	−0.362550
$118$	0	0
$119$	780.985	0.601620
$120$	0	0
$121$	277.420	0.208430
$122$	0	0
$123$	445.316	0.326445
$124$	0	0
$125$	−5386.32	−3.85414
$126$	0	0
$127$	341.377	0.238522	0.119261	−	0.992863i	$-0.461947\pi$
$127$	341.377	0.238522	0.119261	+	0.992863i	$0.461947\pi$
$128$	0	0
$129$	−1914.18	−1.30647
$130$	0	0
$131$	405.014	0.270124	0.135062	−	0.990837i	$-0.456877\pi$
$131$	405.014	0.270124	0.135062	+	0.990837i	$0.456877\pi$
$132$	0	0
$133$	824.627	0.537626
$134$	0	0
$135$	795.645	0.507246
$136$	0	0
$137$	1590.40	0.991803	0.495902	−	0.868379i	$-0.334838\pi$
$137$	1590.40	0.991803	0.495902	+	0.868379i	$0.334838\pi$
$138$	0	0
$139$	605.609	0.369547	0.184774	−	0.982781i	$-0.440845\pi$
$139$	605.609	0.369547	0.184774	+	0.982781i	$0.440845\pi$
$140$	0	0
$141$	−780.677	−0.466276
$142$	0	0
$143$	580.856	0.339676
$144$	0	0
$145$	2061.79	1.18085
$146$	0	0
$147$	−375.352	−0.210602
$148$	0	0
$149$	−1986.22	−1.09206	−0.546032	−	0.837764i	$-0.683862\pi$
$149$	−1986.22	−1.09206	−0.546032	+	0.837764i	$0.683862\pi$
$150$	0	0
$151$	678.905	0.365884	0.182942	−	0.983124i	$-0.441438\pi$
$151$	678.905	0.365884	0.182942	+	0.983124i	$0.441438\pi$
$152$	0	0
$153$	−3534.46	−1.86761
$154$	0	0
$155$	4030.14	2.08844
$156$	0	0
$157$	−3202.21	−1.62780	−0.813899	−	0.581006i	$-0.802659\pi$
$157$	−3202.21	−1.62780	−0.813899	+	0.581006i	$0.802659\pi$
$158$	0	0
$159$	4371.69	2.18049
$160$	0	0
$161$	−838.259	−0.410336
$162$	0	0
$163$	340.941	0.163831	0.0819157	−	0.996639i	$-0.473896\pi$
$163$	340.941	0.163831	0.0819157	+	0.996639i	$0.473896\pi$
$164$	0	0
$165$	−6819.00	−3.21732
$166$	0	0
$167$	−3038.90	−1.40812	−0.704062	−	0.710138i	$-0.748633\pi$
$167$	−3038.90	−1.40812	−0.704062	+	0.710138i	$0.748633\pi$
$168$	0	0
$169$	−1987.23	−0.904521
$170$	0	0
$171$	−3731.97	−1.66895
$172$	0	0
$173$	2611.06	1.14749	0.573743	−	0.819035i	$-0.305491\pi$
$173$	2611.06	1.14749	0.573743	+	0.819035i	$0.305491\pi$
$174$	0	0
$175$	−2573.68	−1.11173
$176$	0	0
$177$	−2124.88	−0.902351
$178$	0	0
$179$	2055.56	0.858325	0.429162	−	0.903227i	$-0.358809\pi$
$179$	2055.56	0.858325	0.429162	+	0.903227i	$0.358809\pi$
$180$	0	0
$181$	−1986.95	−0.815961	−0.407981	−	0.912991i	$-0.633767\pi$
$181$	−1986.95	−0.815961	−0.407981	+	0.912991i	$0.633767\pi$
$182$	0	0
$183$	−635.438	−0.256683
$184$	0	0
$185$	−5484.44	−2.17959
$186$	0	0
$187$	4474.50	1.74977
$188$	0	0
$189$	250.923	0.0965711
$190$	0	0
$191$	−441.866	−0.167394	−0.0836971	−	0.996491i	$-0.526673\pi$
$191$	−441.866	−0.167394	−0.0836971	+	0.996491i	$0.526673\pi$
$192$	0	0
$193$	−5059.63	−1.88705	−0.943524	−	0.331304i	$-0.892511\pi$
$193$	−5059.63	−1.88705	−0.943524	+	0.331304i	$0.892511\pi$
$194$	0	0
$195$	−2462.58	−0.904352
$196$	0	0
$197$	−3150.30	−1.13934	−0.569669	−	0.821875i	$-0.692928\pi$
$197$	−3150.30	−1.13934	−0.569669	+	0.821875i	$0.692928\pi$
$198$	0	0
$199$	−539.846	−0.192305	−0.0961525	−	0.995367i	$-0.530654\pi$
$199$	−539.846	−0.192305	−0.0961525	+	0.995367i	$0.530654\pi$
$200$	0	0
$201$	−3048.54	−1.06979
$202$	0	0
$203$	650.228	0.224813
$204$	0	0
$205$	1290.34	0.439614
$206$	0	0
$207$	3793.66	1.27380
$208$	0	0
$209$	4724.54	1.56365
$210$	0	0
$211$	5413.14	1.76614	0.883071	−	0.469239i	$-0.155472\pi$
$211$	5413.14	1.76614	0.883071	+	0.469239i	$0.155472\pi$
$212$	0	0
$213$	−47.6510	−0.0153286
$214$	0	0
$215$	−5546.48	−1.75938
$216$	0	0
$217$	1270.98	0.397604
$218$	0	0
$219$	−1606.88	−0.495814
$220$	0	0
$221$	1615.89	0.491841
$222$	0	0
$223$	−2289.18	−0.687422	−0.343711	−	0.939075i	$-0.611684\pi$
$223$	−2289.18	−0.687422	−0.343711	+	0.939075i	$0.611684\pi$
$224$	0	0
$225$	11647.6	3.45113
$226$	0	0
$227$	−6742.24	−1.97136	−0.985679	−	0.168634i	$-0.946064\pi$
$227$	−6742.24	−1.97136	−0.985679	+	0.168634i	$0.946064\pi$
$228$	0	0
$229$	1536.97	0.443520	0.221760	−	0.975101i	$-0.428820\pi$
$229$	1536.97	0.443520	0.221760	+	0.975101i	$0.428820\pi$
$230$	0	0
$231$	−2150.51	−0.612524
$232$	0	0
$233$	1568.19	0.440926	0.220463	−	0.975395i	$-0.429243\pi$
$233$	1568.19	0.440926	0.220463	+	0.975395i	$0.429243\pi$
$234$	0	0
$235$	−2262.07	−0.627920
$236$	0	0
$237$	−2082.84	−0.570865
$238$	0	0
$239$	−4907.22	−1.32812	−0.664062	−	0.747677i	$-0.731169\pi$
$239$	−4907.22	−1.32812	−0.664062	+	0.747677i	$0.731169\pi$
$240$	0	0
$241$	1403.57	0.375154	0.187577	−	0.982250i	$-0.439937\pi$
$241$	1403.57	0.375154	0.187577	+	0.982250i	$0.439937\pi$
$242$	0	0
$243$	5416.58	1.42993
$244$	0	0
$245$	−1087.61	−0.283612
$246$	0	0
$247$	1706.19	0.439524
$248$	0	0
$249$	−7739.01	−1.96964
$250$	0	0
$251$	3774.48	0.949175	0.474588	−	0.880208i	$-0.342597\pi$
$251$	3774.48	0.949175	0.474588	+	0.880208i	$0.342597\pi$
$252$	0	0
$253$	−4802.64	−1.19344
$254$	0	0
$255$	−18969.9	−4.65860
$256$	0	0
$257$	−3233.01	−0.784708	−0.392354	−	0.919814i	$-0.628339\pi$
$257$	−3233.01	−0.784708	−0.392354	+	0.919814i	$0.628339\pi$
$258$	0	0
$259$	−1729.63	−0.414957
$260$	0	0
$261$	−2942.70	−0.697887
$262$	0	0
$263$	3259.84	0.764298	0.382149	−	0.924101i	$-0.375184\pi$
$263$	3259.84	0.764298	0.382149	+	0.924101i	$0.375184\pi$
$264$	0	0
$265$	12667.3	2.93640
$266$	0	0
$267$	−5595.21	−1.28248
$268$	0	0
$269$	7328.77	1.66113	0.830563	−	0.556925i	$-0.188019\pi$
$269$	7328.77	1.66113	0.830563	+	0.556925i	$0.188019\pi$
$270$	0	0
$271$	7009.29	1.57116	0.785579	−	0.618761i	$-0.212365\pi$
$271$	7009.29	1.57116	0.785579	+	0.618761i	$0.212365\pi$
$272$	0	0
$273$	−776.622	−0.172173
$274$	0	0
$275$	−14745.4	−3.23339
$276$	0	0
$277$	−3753.77	−0.814233	−0.407116	−	0.913376i	$-0.633466\pi$
$277$	−3753.77	−0.814233	−0.407116	+	0.913376i	$0.633466\pi$
$278$	0	0
$279$	−5752.02	−1.23428
$280$	0	0
$281$	36.9750	0.00784961	0.00392481	−	0.999992i	$-0.498751\pi$
$281$	36.9750	0.00784961	0.00392481	+	0.999992i	$0.498751\pi$
$282$	0	0
$283$	3236.79	0.679884	0.339942	−	0.940446i	$-0.389593\pi$
$283$	3236.79	0.679884	0.339942	+	0.940446i	$0.389593\pi$
$284$	0	0
$285$	−20030.0	−4.16306
$286$	0	0
$287$	406.933	0.0836951
$288$	0	0
$289$	7534.69	1.53362
$290$	0	0
$291$	−5419.25	−1.09169
$292$	0	0
$293$	4412.89	0.879876	0.439938	−	0.898028i	$-0.355000\pi$
$293$	4412.89	0.879876	0.439938	+	0.898028i	$0.355000\pi$
$294$	0	0
$295$	−6157.01	−1.21517
$296$	0	0
$297$	1437.61	0.280871
$298$	0	0
$299$	−1734.40	−0.335461
$300$	0	0
$301$	−1749.19	−0.334956
$302$	0	0
$303$	4569.78	0.866426
$304$	0	0
$305$	−1841.23	−0.345667
$306$	0	0
$307$	588.428	0.109392	0.0546961	−	0.998503i	$-0.482581\pi$
$307$	588.428	0.109392	0.0546961	+	0.998503i	$0.482581\pi$
$308$	0	0
$309$	−4136.22	−0.761493
$310$	0	0
$311$	−7495.29	−1.36662	−0.683310	−	0.730128i	$-0.739460\pi$
$311$	−7495.29	−1.36662	−0.683310	+	0.730128i	$0.739460\pi$
$312$	0	0
$313$	6652.87	1.20141	0.600707	−	0.799469i	$-0.294886\pi$
$313$	6652.87	1.20141	0.600707	+	0.799469i	$0.294886\pi$
$314$	0	0
$315$	4922.14	0.880416
$316$	0	0
$317$	−7215.06	−1.27835	−0.639177	−	0.769060i	$-0.720725\pi$
$317$	−7215.06	−1.27835	−0.639177	+	0.769060i	$0.720725\pi$
$318$	0	0
$319$	3725.35	0.653855
$320$	0	0
$321$	−5059.58	−0.879745
$322$	0	0
$323$	13143.3	2.26412
$324$	0	0
$325$	−5325.08	−0.908868
$326$	0	0
$327$	−1793.70	−0.303339
$328$	0	0
$329$	−713.389	−0.119545
$330$	0	0
$331$	−1123.85	−0.186623	−0.0933116	−	0.995637i	$-0.529745\pi$
$331$	−1123.85	−0.186623	−0.0933116	+	0.995637i	$0.529745\pi$
$332$	0	0
$333$	7827.67	1.28815
$334$	0	0
$335$	−8833.38	−1.44065
$336$	0	0
$337$	−735.459	−0.118881	−0.0594407	−	0.998232i	$-0.518932\pi$
$337$	−735.459	−0.118881	−0.0594407	+	0.998232i	$0.518932\pi$
$338$	0	0
$339$	13334.5	2.13637
$340$	0	0
$341$	7281.85	1.15641
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	20361.1	3.17740
$346$	0	0
$347$	8412.83	1.30151	0.650756	−	0.759287i	$-0.274452\pi$
$347$	8412.83	1.30151	0.650756	+	0.759287i	$0.274452\pi$
$348$	0	0
$349$	8577.90	1.31566	0.657829	−	0.753167i	$-0.271475\pi$
$349$	8577.90	1.31566	0.657829	+	0.753167i	$0.271475\pi$
$350$	0	0
$351$	519.171	0.0789496
$352$	0	0
$353$	−34.2817	−0.00516892	−0.00258446	−	0.999997i	$-0.500823\pi$
$353$	−34.2817	−0.00516892	−0.00258446	+	0.999997i	$0.500823\pi$
$354$	0	0
$355$	−138.072	−0.0206426
$356$	0	0
$357$	−5982.54	−0.886918
$358$	0	0
$359$	1007.89	0.148174	0.0740872	−	0.997252i	$-0.476396\pi$
$359$	1007.89	0.148174	0.0740872	+	0.997252i	$0.476396\pi$
$360$	0	0
$361$	7018.75	1.02329
$362$	0	0
$363$	−2125.11	−0.307271
$364$	0	0
$365$	−4656.07	−0.667698
$366$	0	0
$367$	13076.7	1.85994	0.929969	−	0.367638i	$-0.119834\pi$
$367$	13076.7	1.85994	0.929969	+	0.367638i	$0.119834\pi$
$368$	0	0
$369$	−1841.63	−0.259815
$370$	0	0
$371$	3994.88	0.559040
$372$	0	0
$373$	−8552.62	−1.18723	−0.593616	−	0.804748i	$-0.702300\pi$
$373$	−8552.62	−1.18723	−0.593616	+	0.804748i	$0.702300\pi$
$374$	0	0
$375$	41260.6	5.68183
$376$	0	0
$377$	1345.35	0.183791
$378$	0	0
$379$	−9753.81	−1.32195	−0.660975	−	0.750408i	$-0.729857\pi$
$379$	−9753.81	−1.32195	−0.660975	+	0.750408i	$0.729857\pi$
$380$	0	0
$381$	−2615.03	−0.351633
$382$	0	0
$383$	7141.71	0.952805	0.476402	−	0.879227i	$-0.341941\pi$
$383$	7141.71	0.952805	0.476402	+	0.879227i	$0.341941\pi$
$384$	0	0
$385$	−6231.26	−0.824868
$386$	0	0
$387$	7916.22	1.03980
$388$	0	0
$389$	4854.81	0.632773	0.316386	−	0.948630i	$-0.397530\pi$
$389$	4854.81	0.632773	0.316386	+	0.948630i	$0.397530\pi$
$390$	0	0
$391$	−13360.6	−1.72806
$392$	0	0
$393$	−3102.51	−0.398221
$394$	0	0
$395$	−6035.19	−0.768768
$396$	0	0
$397$	−10401.6	−1.31496	−0.657481	−	0.753471i	$-0.728378\pi$
$397$	−10401.6	−1.31496	−0.657481	+	0.753471i	$0.728378\pi$
$398$	0	0
$399$	−6316.85	−0.792577
$400$	0	0
$401$	4386.56	0.546271	0.273135	−	0.961976i	$-0.411939\pi$
$401$	4386.56	0.546271	0.273135	+	0.961976i	$0.411939\pi$
$402$	0	0
$403$	2629.73	0.325052
$404$	0	0
$405$	12890.6	1.58157
$406$	0	0
$407$	−9909.56	−1.20688
$408$	0	0
$409$	6474.49	0.782746	0.391373	−	0.920232i	$-0.372000\pi$
$409$	6474.49	0.782746	0.391373	+	0.920232i	$0.372000\pi$
$410$	0	0
$411$	−12182.9	−1.46213
$412$	0	0
$413$	−1941.74	−0.231348
$414$	0	0
$415$	−22424.3	−2.65245
$416$	0	0
$417$	−4639.12	−0.544793
$418$	0	0
$419$	−4781.90	−0.557544	−0.278772	−	0.960357i	$-0.589927\pi$
$419$	−4781.90	−0.557544	−0.278772	+	0.960357i	$0.589927\pi$
$420$	0	0
$421$	271.468	0.0314265	0.0157133	−	0.999877i	$-0.494998\pi$
$421$	271.468	0.0314265	0.0157133	+	0.999877i	$0.494998\pi$
$422$	0	0
$423$	3228.54	0.371104
$424$	0	0
$425$	−41020.6	−4.68186
$426$	0	0
$427$	−580.668	−0.0658091
$428$	0	0
$429$	−4449.50	−0.500756
$430$	0	0
$431$	−10024.3	−1.12031	−0.560153	−	0.828389i	$-0.689258\pi$
$431$	−10024.3	−1.12031	−0.560153	+	0.828389i	$0.689258\pi$
$432$	0	0
$433$	14997.2	1.66448	0.832242	−	0.554413i	$-0.187057\pi$
$433$	14997.2	1.66448	0.832242	+	0.554413i	$0.187057\pi$
$434$	0	0
$435$	−15793.9	−1.74082
$436$	0	0
$437$	−14107.2	−1.54425
$438$	0	0
$439$	7619.62	0.828393	0.414196	−	0.910188i	$-0.364063\pi$
$439$	7619.62	0.828393	0.414196	+	0.910188i	$0.364063\pi$
$440$	0	0
$441$	1552.30	0.167616
$442$	0	0
$443$	3188.70	0.341985	0.170993	−	0.985272i	$-0.445303\pi$
$443$	3188.70	0.341985	0.170993	+	0.985272i	$0.445303\pi$
$444$	0	0
$445$	−16212.5	−1.72707
$446$	0	0
$447$	15215.0	1.60994
$448$	0	0
$449$	3446.04	0.362202	0.181101	−	0.983464i	$-0.442034\pi$
$449$	3446.04	0.362202	0.181101	+	0.983464i	$0.442034\pi$
$450$	0	0
$451$	2331.44	0.243422
$452$	0	0
$453$	−5200.58	−0.539392
$454$	0	0
$455$	−2250.32	−0.231861
$456$	0	0
$457$	7559.50	0.773782	0.386891	−	0.922126i	$-0.373549\pi$
$457$	7559.50	0.773782	0.386891	+	0.922126i	$0.373549\pi$
$458$	0	0
$459$	3999.32	0.406694
$460$	0	0
$461$	10358.3	1.04650	0.523249	−	0.852180i	$-0.324720\pi$
$461$	10358.3	1.04650	0.523249	+	0.852180i	$0.324720\pi$
$462$	0	0
$463$	−5546.46	−0.556730	−0.278365	−	0.960475i	$-0.589792\pi$
$463$	−5546.46	−0.556730	−0.278365	+	0.960475i	$0.589792\pi$
$464$	0	0
$465$	−30871.9	−3.07882
$466$	0	0
$467$	−7296.87	−0.723038	−0.361519	−	0.932365i	$-0.617742\pi$
$467$	−7296.87	−0.723038	−0.361519	+	0.932365i	$0.617742\pi$
$468$	0	0
$469$	−2785.78	−0.274276
$470$	0	0
$471$	24529.8	2.39973
$472$	0	0
$473$	−10021.7	−0.974199
$474$	0	0
$475$	−43312.8	−4.18385
$476$	0	0
$477$	−18079.4	−1.73543
$478$	0	0
$479$	−19437.2	−1.85409	−0.927046	−	0.374946i	$-0.877661\pi$
$479$	−19437.2	−1.85409	−0.927046	+	0.374946i	$0.877661\pi$
$480$	0	0
$481$	−3578.68	−0.339239
$482$	0	0
$483$	6421.28	0.604924
$484$	0	0
$485$	−15702.7	−1.47015
$486$	0	0
$487$	6540.19	0.608551	0.304276	−	0.952584i	$-0.401586\pi$
$487$	6540.19	0.608551	0.304276	+	0.952584i	$0.401586\pi$
$488$	0	0
$489$	−2611.69	−0.241523
$490$	0	0
$491$	8542.59	0.785176	0.392588	−	0.919714i	$-0.371580\pi$
$491$	8542.59	0.785176	0.392588	+	0.919714i	$0.371580\pi$
$492$	0	0
$493$	10363.6	0.946764
$494$	0	0
$495$	28200.4	2.56064
$496$	0	0
$497$	−43.5438	−0.00392999
$498$	0	0
$499$	17254.6	1.54794	0.773968	−	0.633224i	$-0.218269\pi$
$499$	17254.6	1.54794	0.773968	+	0.633224i	$0.218269\pi$
$500$	0	0
$501$	23278.7	2.07588
$502$	0	0
$503$	−5597.51	−0.496184	−0.248092	−	0.968736i	$-0.579804\pi$
$503$	−5597.51	−0.496184	−0.248092	+	0.968736i	$0.579804\pi$
$504$	0	0
$505$	13241.3	1.16679
$506$	0	0
$507$	15222.7	1.33346
$508$	0	0
$509$	−5232.92	−0.455688	−0.227844	−	0.973698i	$-0.573168\pi$
$509$	−5232.92	−0.455688	−0.227844	+	0.973698i	$0.573168\pi$
$510$	0	0
$511$	−1468.38	−0.127118
$512$	0	0
$513$	4222.81	0.363434
$514$	0	0
$515$	−11985.0	−1.02548
$516$	0	0
$517$	−4087.22	−0.347690
$518$	0	0
$519$	−20001.4	−1.69164
$520$	0	0
$521$	−7429.89	−0.624778	−0.312389	−	0.949954i	$-0.601129\pi$
$521$	−7429.89	−0.624778	−0.312389	+	0.949954i	$0.601129\pi$
$522$	0	0
$523$	−2963.01	−0.247732	−0.123866	−	0.992299i	$-0.539529\pi$
$523$	−2963.01	−0.247732	−0.123866	+	0.992299i	$0.539529\pi$
$524$	0	0
$525$	19715.1	1.63893
$526$	0	0
$527$	20257.5	1.67444
$528$	0	0
$529$	2173.37	0.178628
$530$	0	0
$531$	8787.60	0.718172
$532$	0	0
$533$	841.964	0.0684231
$534$	0	0
$535$	−14660.5	−1.18473
$536$	0	0
$537$	−15746.1	−1.26536
$538$	0	0
$539$	−1965.15	−0.157041
$540$	0	0
$541$	19355.9	1.53822	0.769109	−	0.639118i	$-0.220701\pi$
$541$	19355.9	1.53822	0.769109	+	0.639118i	$0.220701\pi$
$542$	0	0
$543$	15220.6	1.20290
$544$	0	0
$545$	−5197.38	−0.408498
$546$	0	0
$547$	−7653.95	−0.598280	−0.299140	−	0.954209i	$-0.596700\pi$
$547$	−7653.95	−0.598280	−0.299140	+	0.954209i	$0.596700\pi$
$548$	0	0
$549$	2627.90	0.204291
$550$	0	0
$551$	10942.8	0.846057
$552$	0	0
$553$	−1903.32	−0.146360
$554$	0	0
$555$	42012.2	3.21319
$556$	0	0
$557$	10896.0	0.828864	0.414432	−	0.910080i	$-0.363980\pi$
$557$	10896.0	0.828864	0.414432	+	0.910080i	$0.363980\pi$
$558$	0	0
$559$	−3619.16	−0.273836
$560$	0	0
$561$	−34275.8	−2.57954
$562$	0	0
$563$	−15441.8	−1.15594	−0.577972	−	0.816057i	$-0.696156\pi$
$563$	−15441.8	−1.15594	−0.577972	+	0.816057i	$0.696156\pi$
$564$	0	0
$565$	38637.6	2.87699
$566$	0	0
$567$	4065.29	0.301104
$568$	0	0
$569$	5615.59	0.413739	0.206870	−	0.978369i	$-0.433672\pi$
$569$	5615.59	0.413739	0.206870	+	0.978369i	$0.433672\pi$
$570$	0	0
$571$	14371.0	1.05325	0.526626	−	0.850097i	$-0.323457\pi$
$571$	14371.0	1.05325	0.526626	+	0.850097i	$0.323457\pi$
$572$	0	0
$573$	3384.81	0.246775
$574$	0	0
$575$	44028.9	3.19327
$576$	0	0
$577$	−18933.5	−1.36605	−0.683027	−	0.730393i	$-0.739337\pi$
$577$	−18933.5	−1.36605	−0.683027	+	0.730393i	$0.739337\pi$
$578$	0	0
$579$	38758.1	2.78192
$580$	0	0
$581$	−7071.96	−0.504982
$582$	0	0
$583$	22887.9	1.62593
$584$	0	0
$585$	10184.1	0.719765
$586$	0	0
$587$	17871.6	1.25663	0.628315	−	0.777959i	$-0.283745\pi$
$587$	17871.6	1.25663	0.628315	+	0.777959i	$0.283745\pi$
$588$	0	0
$589$	21389.6	1.49633
$590$	0	0
$591$	24132.1	1.67963
$592$	0	0
$593$	45.4735	0.00314902	0.00157451	−	0.999999i	$-0.499499\pi$
$593$	45.4735	0.00314902	0.00157451	+	0.999999i	$0.499499\pi$
$594$	0	0
$595$	−17334.9	−1.19439
$596$	0	0
$597$	4135.36	0.283499
$598$	0	0
$599$	10336.6	0.705078	0.352539	−	0.935797i	$-0.385318\pi$
$599$	10336.6	0.705078	0.352539	+	0.935797i	$0.385318\pi$
$600$	0	0
$601$	−4844.95	−0.328835	−0.164417	−	0.986391i	$-0.552574\pi$
$601$	−4844.95	−0.328835	−0.164417	+	0.986391i	$0.552574\pi$
$602$	0	0
$603$	12607.5	0.851435
$604$	0	0
$605$	−6157.67	−0.413793
$606$	0	0
$607$	−15710.1	−1.05050	−0.525249	−	0.850949i	$-0.676028\pi$
$607$	−15710.1	−1.05050	−0.525249	+	0.850949i	$0.676028\pi$
$608$	0	0
$609$	−4980.91	−0.331423
$610$	0	0
$611$	−1476.04	−0.0977317
$612$	0	0
$613$	−23320.2	−1.53653	−0.768267	−	0.640130i	$-0.778881\pi$
$613$	−23320.2	−1.53653	−0.768267	+	0.640130i	$0.778881\pi$
$614$	0	0
$615$	−9884.30	−0.648087
$616$	0	0
$617$	10540.0	0.687720	0.343860	−	0.939021i	$-0.388266\pi$
$617$	10540.0	0.687720	0.343860	+	0.939021i	$0.388266\pi$
$618$	0	0
$619$	17788.6	1.15507	0.577533	−	0.816367i	$-0.304016\pi$
$619$	17788.6	1.15507	0.577533	+	0.816367i	$0.304016\pi$
$620$	0	0
$621$	−4292.62	−0.277386
$622$	0	0
$623$	−5112.94	−0.328805
$624$	0	0
$625$	73597.0	4.71021
$626$	0	0
$627$	−36191.2	−2.30516
$628$	0	0
$629$	−27567.6	−1.74752
$630$	0	0
$631$	12278.6	0.774648	0.387324	−	0.921944i	$-0.373399\pi$
$631$	12278.6	0.774648	0.387324	+	0.921944i	$0.373399\pi$
$632$	0	0
$633$	−41466.1	−2.60368
$634$	0	0
$635$	−7577.25	−0.473534
$636$	0	0
$637$	−709.684	−0.0441424
$638$	0	0
$639$	197.064	0.0121999
$640$	0	0
$641$	15729.2	0.969216	0.484608	−	0.874732i	$-0.338962\pi$
$641$	15729.2	0.969216	0.484608	+	0.874732i	$0.338962\pi$
$642$	0	0
$643$	−7527.17	−0.461652	−0.230826	−	0.972995i	$-0.574143\pi$
$643$	−7527.17	−0.461652	−0.230826	+	0.972995i	$0.574143\pi$
$644$	0	0
$645$	42487.4	2.59371
$646$	0	0
$647$	−4431.02	−0.269245	−0.134622	−	0.990897i	$-0.542982\pi$
$647$	−4431.02	−0.269245	−0.134622	+	0.990897i	$0.542982\pi$
$648$	0	0
$649$	−11124.8	−0.672860
$650$	0	0
$651$	−9736.06	−0.586154
$652$	0	0
$653$	−4155.52	−0.249032	−0.124516	−	0.992218i	$-0.539738\pi$
$653$	−4155.52	−0.249032	−0.124516	+	0.992218i	$0.539738\pi$
$654$	0	0
$655$	−8989.75	−0.536272
$656$	0	0
$657$	6645.38	0.394613
$658$	0	0
$659$	10380.1	0.613583	0.306791	−	0.951777i	$-0.400745\pi$
$659$	10380.1	0.613583	0.306791	+	0.951777i	$0.400745\pi$
$660$	0	0
$661$	17788.9	1.04676	0.523381	−	0.852099i	$-0.324671\pi$
$661$	17788.9	1.04676	0.523381	+	0.852099i	$0.324671\pi$
$662$	0	0
$663$	−12378.2	−0.725080
$664$	0	0
$665$	−18303.5	−1.06734
$666$	0	0
$667$	−11123.7	−0.645742
$668$	0	0
$669$	17535.7	1.01341
$670$	0	0
$671$	−3326.82	−0.191402
$672$	0	0
$673$	13619.4	0.780074	0.390037	−	0.920799i	$-0.372462\pi$
$673$	13619.4	0.780074	0.390037	+	0.920799i	$0.372462\pi$
$674$	0	0
$675$	−13179.5	−0.751525
$676$	0	0
$677$	10454.1	0.593478	0.296739	−	0.954959i	$-0.404101\pi$
$677$	10454.1	0.593478	0.296739	+	0.954959i	$0.404101\pi$
$678$	0	0
$679$	−4952.16	−0.279892
$680$	0	0
$681$	51647.3	2.90621
$682$	0	0
$683$	−11405.9	−0.638996	−0.319498	−	0.947587i	$-0.603514\pi$
$683$	−11405.9	−0.638996	−0.319498	+	0.947587i	$0.603514\pi$
$684$	0	0
$685$	−35300.8	−1.96901
$686$	0	0
$687$	−11773.6	−0.653844
$688$	0	0
$689$	8265.60	0.457031
$690$	0	0
$691$	5921.64	0.326006	0.163003	−	0.986626i	$-0.447882\pi$
$691$	5921.64	0.326006	0.163003	+	0.986626i	$0.447882\pi$
$692$	0	0
$693$	8893.57	0.487502
$694$	0	0
$695$	−13442.2	−0.733656
$696$	0	0
$697$	6485.89	0.352468
$698$	0	0
$699$	−12012.8	−0.650021
$700$	0	0
$701$	24236.4	1.30584	0.652921	−	0.757426i	$-0.273543\pi$
$701$	24236.4	1.30584	0.652921	+	0.757426i	$0.273543\pi$
$702$	0	0
$703$	−29108.1	−1.56164
$704$	0	0
$705$	17328.0	0.925690
$706$	0	0
$707$	4175.90	0.222137
$708$	0	0
$709$	30145.3	1.59680	0.798400	−	0.602128i	$-0.205680\pi$
$709$	30145.3	1.59680	0.798400	+	0.602128i	$0.205680\pi$
$710$	0	0
$711$	8613.73	0.454346
$712$	0	0
$713$	−21743.1	−1.14206
$714$	0	0
$715$	−12892.8	−0.674353
$716$	0	0
$717$	37590.6	1.95794
$718$	0	0
$719$	5159.52	0.267619	0.133809	−	0.991007i	$-0.457279\pi$
$719$	5159.52	0.267619	0.133809	+	0.991007i	$0.457279\pi$
$720$	0	0
$721$	−3779.71	−0.195234
$722$	0	0
$723$	−10751.7	−0.553059
$724$	0	0
$725$	−34152.7	−1.74952
$726$	0	0
$727$	22475.2	1.14657	0.573286	−	0.819355i	$-0.305668\pi$
$727$	22475.2	1.14657	0.573286	+	0.819355i	$0.305668\pi$
$728$	0	0
$729$	−25812.0	−1.31139
$730$	0	0
$731$	−27879.4	−1.41061
$732$	0	0
$733$	15258.4	0.768871	0.384436	−	0.923152i	$-0.374396\pi$
$733$	15258.4	0.768871	0.384436	+	0.923152i	$0.374396\pi$
$734$	0	0
$735$	8331.38	0.418106
$736$	0	0
$737$	−15960.6	−0.797715
$738$	0	0
$739$	−27519.3	−1.36984	−0.684922	−	0.728616i	$-0.740164\pi$
$739$	−27519.3	−1.36984	−0.684922	+	0.728616i	$0.740164\pi$
$740$	0	0
$741$	−13069.9	−0.647954
$742$	0	0
$743$	−38497.6	−1.90086	−0.950431	−	0.310937i	$-0.899357\pi$
$743$	−38497.6	−1.90086	−0.950431	+	0.310937i	$0.899357\pi$
$744$	0	0
$745$	44086.5	2.16806
$746$	0	0
$747$	32005.2	1.56761
$748$	0	0
$749$	−4623.48	−0.225552
$750$	0	0
$751$	−25060.1	−1.21765	−0.608827	−	0.793303i	$-0.708359\pi$
$751$	−25060.1	−1.21765	−0.608827	+	0.793303i	$0.708359\pi$
$752$	0	0
$753$	−28913.5	−1.39929
$754$	0	0
$755$	−15069.1	−0.726384
$756$	0	0
$757$	7440.21	0.357225	0.178612	−	0.983920i	$-0.442839\pi$
$757$	7440.21	0.357225	0.178612	+	0.983920i	$0.442839\pi$
$758$	0	0
$759$	36789.4	1.75938
$760$	0	0
$761$	−11058.6	−0.526773	−0.263386	−	0.964690i	$-0.584839\pi$
$761$	−11058.6	−0.526773	−0.263386	+	0.964690i	$0.584839\pi$
$762$	0	0
$763$	−1639.10	−0.0777710
$764$	0	0
$765$	78451.3	3.70773
$766$	0	0
$767$	−4017.55	−0.189133
$768$	0	0
$769$	39340.3	1.84479	0.922397	−	0.386243i	$-0.126227\pi$
$769$	39340.3	1.84479	0.922397	+	0.386243i	$0.126227\pi$
$770$	0	0
$771$	24765.7	1.15683
$772$	0	0
$773$	−28953.3	−1.34719	−0.673595	−	0.739100i	$-0.735251\pi$
$773$	−28953.3	−1.34719	−0.673595	+	0.739100i	$0.735251\pi$
$774$	0	0
$775$	−66757.4	−3.09419
$776$	0	0
$777$	13249.4	0.611736
$778$	0	0
$779$	6848.32	0.314976
$780$	0	0
$781$	−249.476	−0.0114301
$782$	0	0
$783$	3329.73	0.151973
$784$	0	0
$785$	71076.8	3.23164
$786$	0	0
$787$	15424.4	0.698627	0.349313	−	0.937006i	$-0.386415\pi$
$787$	15424.4	0.698627	0.349313	+	0.937006i	$0.386415\pi$
$788$	0	0
$789$	−24971.2	−1.12674
$790$	0	0
$791$	12185.2	0.547730
$792$	0	0
$793$	−1201.43	−0.0538008
$794$	0	0
$795$	−97034.6	−4.32888
$796$	0	0
$797$	−13348.5	−0.593258	−0.296629	−	0.954993i	$-0.595862\pi$
$797$	−13348.5	−0.593258	−0.296629	+	0.954993i	$0.595862\pi$
$798$	0	0
$799$	−11370.3	−0.503446
$800$	0	0
$801$	23139.3	1.02071
$802$	0	0
$803$	−8412.82	−0.369716
$804$	0	0
$805$	18606.1	0.814633
$806$	0	0
$807$	−56140.2	−2.44886
$808$	0	0
$809$	−18871.7	−0.820139	−0.410070	−	0.912054i	$-0.634496\pi$
$809$	−18871.7	−0.820139	−0.410070	+	0.912054i	$0.634496\pi$
$810$	0	0
$811$	8113.79	0.351312	0.175656	−	0.984452i	$-0.443795\pi$
$811$	8113.79	0.351312	0.175656	+	0.984452i	$0.443795\pi$
$812$	0	0
$813$	−53692.9	−2.31623
$814$	0	0
$815$	−7567.57	−0.325252
$816$	0	0
$817$	−29437.4	−1.26057
$818$	0	0
$819$	3211.77	0.137031
$820$	0	0
$821$	16368.2	0.695804	0.347902	−	0.937531i	$-0.386894\pi$
$821$	16368.2	0.695804	0.347902	+	0.937531i	$0.386894\pi$
$822$	0	0
$823$	−34575.7	−1.46444	−0.732220	−	0.681069i	$-0.761516\pi$
$823$	−34575.7	−1.46444	−0.732220	+	0.681069i	$0.761516\pi$
$824$	0	0
$825$	112954.	4.76672
$826$	0	0
$827$	43222.0	1.81739	0.908693	−	0.417466i	$-0.137082\pi$
$827$	43222.0	1.81739	0.908693	+	0.417466i	$0.137082\pi$
$828$	0	0
$829$	−29288.6	−1.22706	−0.613532	−	0.789670i	$-0.710252\pi$
$829$	−29288.6	−1.22706	−0.613532	+	0.789670i	$0.710252\pi$
$830$	0	0
$831$	28754.9	1.20036
$832$	0	0
$833$	−5466.89	−0.227391
$834$	0	0
$835$	67451.8	2.79553
$836$	0	0
$837$	6508.55	0.268779
$838$	0	0
$839$	−16518.3	−0.679708	−0.339854	−	0.940478i	$-0.610378\pi$
$839$	−16518.3	−0.679708	−0.339854	+	0.940478i	$0.610378\pi$
$840$	0	0
$841$	−15760.5	−0.646213
$842$	0	0
$843$	−283.238	−0.0115720
$844$	0	0
$845$	44108.9	1.79573
$846$	0	0
$847$	−1941.94	−0.0787792
$848$	0	0
$849$	−24794.6	−1.00230
$850$	0	0
$851$	29589.3	1.19190
$852$	0	0
$853$	8649.24	0.347180	0.173590	−	0.984818i	$-0.444463\pi$
$853$	8649.24	0.347180	0.173590	+	0.984818i	$0.444463\pi$
$854$	0	0
$855$	82835.3	3.31334
$856$	0	0
$857$	−16342.9	−0.651413	−0.325707	−	0.945471i	$-0.605602\pi$
$857$	−16342.9	−0.651413	−0.325707	+	0.945471i	$0.605602\pi$
$858$	0	0
$859$	29613.4	1.17625	0.588123	−	0.808772i	$-0.299867\pi$
$859$	29613.4	1.17625	0.588123	+	0.808772i	$0.299867\pi$
$860$	0	0
$861$	−3117.21	−0.123385
$862$	0	0
$863$	−20366.7	−0.803348	−0.401674	−	0.915783i	$-0.631571\pi$
$863$	−20366.7	−0.803348	−0.401674	+	0.915783i	$0.631571\pi$
$864$	0	0
$865$	−57955.5	−2.27809
$866$	0	0
$867$	−57717.6	−2.26089
$868$	0	0
$869$	−10904.7	−0.425680
$870$	0	0
$871$	−5763.92	−0.224228
$872$	0	0
$873$	22411.7	0.868867
$874$	0	0
$875$	37704.3	1.45673
$876$	0	0
$877$	−35210.6	−1.35573	−0.677866	−	0.735186i	$-0.737095\pi$
$877$	−35210.6	−1.35573	−0.677866	+	0.735186i	$0.737095\pi$
$878$	0	0
$879$	−33803.8	−1.29713
$880$	0	0
$881$	−7354.03	−0.281230	−0.140615	−	0.990064i	$-0.544908\pi$
$881$	−7354.03	−0.281230	−0.140615	+	0.990064i	$0.544908\pi$
$882$	0	0
$883$	28837.9	1.09906	0.549531	−	0.835473i	$-0.314806\pi$
$883$	28837.9	1.09906	0.549531	+	0.835473i	$0.314806\pi$
$884$	0	0
$885$	47164.3	1.79142
$886$	0	0
$887$	10861.0	0.411136	0.205568	−	0.978643i	$-0.434096\pi$
$887$	10861.0	0.411136	0.205568	+	0.978643i	$0.434096\pi$
$888$	0	0
$889$	−2389.64	−0.0901528
$890$	0	0
$891$	23291.3	0.875744
$892$	0	0
$893$	−12005.7	−0.449895
$894$	0	0
$895$	−45625.6	−1.70402
$896$	0	0
$897$	13285.9	0.494542
$898$	0	0
$899$	16865.9	0.625706
$900$	0	0
$901$	63672.3	2.35431
$902$	0	0
$903$	13399.3	0.493798
$904$	0	0
$905$	44102.7	1.61992
$906$	0	0
$907$	18682.8	0.683961	0.341980	−	0.939707i	$-0.388902\pi$
$907$	18682.8	0.683961	0.341980	+	0.939707i	$0.388902\pi$
$908$	0	0
$909$	−18898.6	−0.689580
$910$	0	0
$911$	−34392.2	−1.25078	−0.625392	−	0.780311i	$-0.715061\pi$
$911$	−34392.2	−1.25078	−0.625392	+	0.780311i	$0.715061\pi$
$912$	0	0
$913$	−40517.4	−1.46871
$914$	0	0
$915$	14104.3	0.509588
$916$	0	0
$917$	−2835.10	−0.102097
$918$	0	0
$919$	−14715.8	−0.528216	−0.264108	−	0.964493i	$-0.585078\pi$
$919$	−14715.8	−0.528216	−0.264108	+	0.964493i	$0.585078\pi$
$920$	0	0
$921$	−4507.51	−0.161268
$922$	0	0
$923$	−90.0943	−0.00321288
$924$	0	0
$925$	90847.3	3.22923
$926$	0	0
$927$	17105.6	0.606065
$928$	0	0
$929$	−36204.2	−1.27860	−0.639300	−	0.768957i	$-0.720776\pi$
$929$	−36204.2	−1.27860	−0.639300	+	0.768957i	$0.720776\pi$
$930$	0	0
$931$	−5772.39	−0.203203
$932$	0	0
$933$	57415.8	2.01469
$934$	0	0
$935$	−99316.6	−3.47380
$936$	0	0
$937$	−43636.5	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$937$	−43636.5	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$938$	0	0
$939$	−50962.7	−1.77114
$940$	0	0
$941$	−4751.54	−0.164608	−0.0823038	−	0.996607i	$-0.526228\pi$
$941$	−4751.54	−0.164608	−0.0823038	+	0.996607i	$0.526228\pi$
$942$	0	0
$943$	−6961.54	−0.240402
$944$	0	0
$945$	−5569.52	−0.191721
$946$	0	0
$947$	−49387.3	−1.69469	−0.847344	−	0.531044i	$-0.821800\pi$
$947$	−49387.3	−1.69469	−0.847344	+	0.531044i	$0.821800\pi$
$948$	0	0
$949$	−3038.16	−0.103923
$950$	0	0
$951$	55269.2	1.88457
$952$	0	0
$953$	46336.1	1.57500	0.787499	−	0.616316i	$-0.211375\pi$
$953$	46336.1	1.57500	0.787499	+	0.616316i	$0.211375\pi$
$954$	0	0
$955$	9807.72	0.332325
$956$	0	0
$957$	−28537.2	−0.963924
$958$	0	0
$959$	−11132.8	−0.374866
$960$	0	0
$961$	3176.38	0.106622
$962$	0	0
$963$	20924.2	0.700180
$964$	0	0
$965$	112304.	3.74633
$966$	0	0
$967$	21078.1	0.700957	0.350479	−	0.936571i	$-0.386019\pi$
$967$	21078.1	0.700957	0.350479	+	0.936571i	$0.386019\pi$
$968$	0	0
$969$	−100681.	−3.33781
$970$	0	0
$971$	−27225.5	−0.899802	−0.449901	−	0.893079i	$-0.648541\pi$
$971$	−27225.5	−0.899802	−0.449901	+	0.893079i	$0.648541\pi$
$972$	0	0
$973$	−4239.26	−0.139676
$974$	0	0
$975$	40791.4	1.33987
$976$	0	0
$977$	4699.41	0.153887	0.0769434	−	0.997035i	$-0.475484\pi$
$977$	4699.41	0.153887	0.0769434	+	0.997035i	$0.475484\pi$
$978$	0	0
$979$	−29293.6	−0.956310
$980$	0	0
$981$	7417.96	0.241424
$982$	0	0
$983$	48128.7	1.56161	0.780807	−	0.624773i	$-0.214808\pi$
$983$	48128.7	1.56161	0.780807	+	0.624773i	$0.214808\pi$
$984$	0	0
$985$	69924.5	2.26191
$986$	0	0
$987$	5464.74	0.176236
$988$	0	0
$989$	29924.0	0.962111
$990$	0	0
$991$	−214.168	−0.00686506	−0.00343253	−	0.999994i	$-0.501093\pi$
$991$	−214.168	−0.00686506	−0.00343253	+	0.999994i	$0.501093\pi$
$992$	0	0
$993$	8608.96	0.275123
$994$	0	0
$995$	11982.5	0.381780
$996$	0	0
$997$	12678.3	0.402735	0.201368	−	0.979516i	$-0.435461\pi$
$997$	12678.3	0.402735	0.201368	+	0.979516i	$0.435461\pi$
$998$	0	0
$999$	−8857.20	−0.280510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.4.a.c.1.1		2
4.3	odd	2		1792.4.a.a.1.2		2
8.3	odd	2		1792.4.a.d.1.1		2
8.5	even	2		1792.4.a.b.1.2		2
16.3	odd	4		448.4.b.a.225.3	yes	4
16.5	even	4		448.4.b.b.225.3	yes	4
16.11	odd	4		448.4.b.a.225.2	✓	4
16.13	even	4		448.4.b.b.225.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.4.b.a.225.2	✓	4	16.11	odd	4
448.4.b.a.225.3	yes	4	16.3	odd	4
448.4.b.b.225.2	yes	4	16.13	even	4
448.4.b.b.225.3	yes	4	16.5	even	4
1792.4.a.a.1.2		2	4.3	odd	2
1792.4.a.b.1.2		2	8.5	even	2
1792.4.a.c.1.1		2	1.1	even	1	trivial
1792.4.a.d.1.1		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1792.a (trivial)

\(f(q)\)	\(=\)	\(q-7.66025 q^{3} -22.1962 q^{5} -7.00000 q^{7} +31.6795 q^{9} -40.1051 q^{11} -14.4833 q^{13} +170.028 q^{15} -111.569 q^{17} -117.804 q^{19} +53.6218 q^{21} +119.751 q^{23} +367.669 q^{25} -35.8461 q^{27} -92.8897 q^{29} -181.569 q^{31} +307.215 q^{33} +155.373 q^{35} +247.090 q^{37} +110.946 q^{39} -58.1333 q^{41} +249.885 q^{43} -703.163 q^{45} +101.913 q^{47} +49.0000 q^{49} +854.649 q^{51} -570.697 q^{53} +890.179 q^{55} +902.407 q^{57} +277.391 q^{59} +82.9526 q^{61} -221.756 q^{63} +321.474 q^{65} +397.969 q^{67} -917.325 q^{69} +6.22055 q^{71} +209.769 q^{73} -2816.44 q^{75} +280.736 q^{77} +271.902 q^{79} -580.756 q^{81} +1010.28 q^{83} +2476.41 q^{85} +711.559 q^{87} +730.420 q^{89} +101.383 q^{91} +1390.87 q^{93} +2614.79 q^{95} +707.451 q^{97} -1270.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 34 q^{5} - 14 q^{7} + 98 q^{9} - 4 q^{11} - 74 q^{13} + 56 q^{15} - 140 q^{17} - 246 q^{19} - 14 q^{21} + 288 q^{23} + 382 q^{25} + 344 q^{27} - 68 q^{29} - 280 q^{31} + 656 q^{33} + 238 q^{35}+ \cdots + 1124 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 34 * q^5 - 14 * q^7 + 98 * q^9 - 4 * q^11 - 74 * q^13 + 56 * q^15 - 140 * q^17 - 246 * q^19 - 14 * q^21 + 288 * q^23 + 382 * q^25 + 344 * q^27 - 68 * q^29 - 280 * q^31 + 656 * q^33 + 238 * q^35 - 164 * q^37 - 464 * q^39 + 244 * q^41 + 188 * q^43 - 1486 * q^45 - 392 * q^47 + 98 * q^49 + 580 * q^51 - 504 * q^53 + 464 * q^55 - 336 * q^57 - 190 * q^59 + 398 * q^61 - 686 * q^63 + 1024 * q^65 - 368 * q^67 + 708 * q^69 + 248 * q^71 - 204 * q^73 - 2678 * q^75 + 28 * q^77 - 440 * q^79 + 1298 * q^81 + 1158 * q^83 + 2812 * q^85 + 952 * q^87 + 1156 * q^89 + 518 * q^91 + 440 * q^93 + 4128 * q^95 - 428 * q^97 + 1124 * q^99

Introduction

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